Htykuut

a) Let $B$ be a 2x2 symmetric matrix and let $u$, $v$ be two eigenvectors of $B$ associated with the eigenvalues $w$ and $l$ respectively. Suppose $u$ = $$ left[
begin{array}{c}
1/sqrt2\
-1/sqrt2
end{array}
right] $$
and $v$ = $$ left[
begin{array}{c}
1/sqrt2\
1/sqrt2
end{array}
right] $$
with $w$= 1 and $l$=3 respectively, find matrix $B$.

b) Let $C$ be another symmetric matrix of order n with a characteristic polynomial $(w-w_1)(w-w_2)...(w-w_n)$ where $w_1≤w_2≤...≤w_n$. Prove that for any nonzero vector $x$ that belongs in $R^n$, $w_1≤x^TCx/x^Tx≤w_n$.

What I have done:
For part (a), converting $u$ and $v$ to orthogonal basis, I get $u_o$=$$ left[
begin{array}{c}
1\
-1
end{array}
right] $$ and $v_o$=$$ left[
begin{array}{c}
1\
1
end{array}
right] $$. Letting $M$ = $$ left[
begin{array}{cc}
1&0\
0&3
end{array}
right] $$ and $S$ = $$ left[
begin{array}{cc}
1&1\
-1&1
end{array}
right] $$, the matrix B is obtained by putting together $SMS^{-1}$ = $$ left[
begin{array}{cc}
2&1\
1&2
end{array}
right] $$.

However, for part (b), while I can derive that $w_1,w_2,...,w_n$ are eigenvalues of C, I am unsure of how to proceed, hence may I get some help for this?

Suppose that $lVert xrVert=1$. Then $x^Tx=1$. On the other hand, let $(v_1,ldots,v_n)$ an orthonormal basis of eigenvecors, such that the eigenvalu corresponding to $v_k$ is $w_k$. Then $x$ can be written as $alpha_1v_1+cdots+alpha_kv_k$. So, $Cx=alpha_1w_1v_1+cdots+alpha_nw_nv_n$ and so$$x^TCX=lvertalpha_1rvert^2w_1+cdots+lvertalpha_nrvert^2w_n.$$Since $lvertalpha_1rvert^2+cdots+lvertalpha_nrvert^2=1$, this number is between $w_1$ and $w_n$.

If $x$ is an arbitrary non-zero vector, you can apply the previus result to $frac x{lVert xrVert}$.